Science & Mathematics

Additive codes generalize linear codes and play an important role in the construc?tion of quantum codes. As generalizations of linear MDS codes, additive MDS codes have received much research attention. As an important subclass of additive codes, ACD codes are widely used for quantum error correction. In this paper, we study single-twist additive generalized twisted Reed-Solomon (GTRS) codes over finite fields. We derive necessary and sufficient conditions for these codes to be MDS for arbitrary hook and twist parameters. Then we employ Schur squares to prove that such additive MDS codes are not monomially equivalent to additive Reed-Solomon codes. We also identify a class of additive GTRS codes whose duals remain additive GTRS codes, and show that, for all sufficiently large prime powers q whose characteristic avoids a finite set, such GTRS codes yield ACD codes.

We present the directed simplicial complex: this is an extension of a simplicial complex with a direction on its faces. This enables us to model a local notion of belief in a three-valued semantics, in which formulas may be true, false, or undefined. Of particular interest is a class of directed simplicial complexes called cups, in which any maximal set of vertices of different colors is a facet. Cups permit a more concise relational representation. We give examples and applications of our logical semantics. We then provide a sound and complete axiomatization for the class of directed simplicial complexes and extend it to an axiomatization for the class of cups. We include a detailed technical comparison with other works on belief on complexes. Our work generalizes a prior axiomatization for knowledge on (undirected) impure simplicial complexes.

Deep neural networks typically rely on architectures specified before training, despite the continuous evolution of their internal representations. We introduce FlowGenesis, a framework in which neural architectures adapt through neuron birth and pruning guided by Representation Flux, a local statistic capturing changes in minibatch mean activations. Neurons exhibiting persistently high Flux undergo splitting designed to approximately preserve the network function, while neurons with low Flux and low activity are pruned. This mechanism allows network capacity to evolve without prescribing a target architecture or performing a separate architecture search. Experiments on image and text classi- fication show that networks initialized with different sizes approach similar final neuron-count ranges within each dataset while maintaining comparable predictive performance. We further extend FlowGenesis to continual learning, where representation-guided structural adaptation reduces the neurons and parameters allocated to successive tasks while maintaining accuracy comparable to fixed-width Progressive Neural Networks in a task-incremental learning setting. These findings demonstrate that neural architectures can adapt through feedback between representation learning and structural change, suggesting an alternative to prescribing network capacity before training.

Einstein relations connect carrier mobility to macroscopic transport coefficients and constitute a fundamental link between microscopic dynamics and transport. In phonon hydrodynamics, this connection is conventionally local: the drift-induced thermal conductivity is proportional to the local drift mobility. Here we show that this relation becomes intrinsically nonlocal under spatial confinement. We derive a nonlocal Einstein relation, $κ_H(y) = \int P(y-y')μ(y'),dy'$, in which hydrodynamic thermal conductivity $κ_H(y)$ at one position depends on collective drift mobility $μ(y')$ across the entire cross-section of the channel. In the small-Knudsen-number limit, our nonlocal theory recovers the viscous heat equation and the local Einstein relation. At finite Knudsen number, nonlocality produces intrinsic boundary slip and distinct size-scaling behavior of the drift mobility $μ$ and hydrodynamic thermal conductivity $κ_H(y)$. First-principles calculations for graphite show that nonlocality broadens the channel-size window over which thermal conductivity exhibits superlinear size dependence. Temperature-dependent simulations further show that the nonlocal theory captures the emergence of a hydrodynamic peak in thermal conductivity normalized by ballistic conductance, whereas the local theory fails to accurately predict its magnitude. Our results establish a microscopic connection between collective phonon mobility and heat conduction beyond local hydrodynamics.

A cooperative 3D detector can miss a vehicle even when the LiDAR points on it remain unchanged. We study this failure under adverse weather as Weather-Induced Context Interference. To our knowledge, this is the first controlled study of how weather-related exterior changes remove cooperative benefits while preserving all points inside annotated objects. Controlled experiments on simulated and real roadside inputs show that exterior loss changes object representations before fusion and removes previously useful assistance. On real inputs, restoring object-neighborhood features recovers 30 of 31 induced failures; three equal-area exterior controls recover none. The failure condition also appears with a second fusion model. These findings motivate Adverse-Weather Cooperative Repair (AWCR), with separate implementations for the available supervision. Point Classifier filters added weather returns in simulation. On real inputs, Local Correction refines candidates and Quality Selection checks them against current measurements, without naturally paired clear/adverse-weather recordings. Candidate tracing shows that correction supplies accurate geometry for some targets that original candidates cannot localize at the evaluation threshold. On natural adverse-weather CoInfra development data, AWCR improves car bird's-eye-view (BEV) [email protected] by 4.66 points with V2X-ViT and 5.56 with AttFuse. On simulated V2X-R fog, it improves vehicle BEV [email protected] by 4.47 points at a fixed 4,096-candidate budget.

Large language models operationalize an intuitive, view of meaning as use. We argue that it faces a rule-following challenge. Using Kripke's interpretation of Wittgenstein, we argue that finite histories of use do not determine a unique rule for future extension. This reframes the debate about OOD generalization but also about NLP evaluation; benchmarks that measure performance on fixed use cases or local robustness, we argue, do not by themselves establish semantic competence. To evaluate whether models preserve rules and concepts under principled domain extension, we propose to evaluate two capacities, central to this ability: abstraction and recursion; and we offer an experimental paradigm for testing both.

Grokking describes understanding something deeply and intuitively. In neural networks, it is the phenomenon where a network generalizes long after memorizing its training data. This delay has been linked to a fall in the effective rank of the network's weight matrices, but whether this rank compression drives grokking or follows it is still unresolved. We study a mean-field MLP and a one-layer transformer trained on modular arithmetic and observe that: (i) Compression follows the onset, not memorization. Every transformer memorizes its training data within 250 steps, while the grokking onset (the first step at which test accuracy reaches 95%) varies from 6,500 to 21,000 steps. The steepest fall in effective rank comes about 900 steps after the onset, however late the onset occurs, and networks that memorize without grokking compress far less. (ii) Subspace crystallization. We call the span of the fewest top singular directions of a weight matrix that preserve 99% of test accuracy its sufficient subspace. Its size can shrink as the margin between the correct and the best wrong output grows, so we also count the directions needed to keep the outputs within 10%. In the transformer, this count falls during the transition from a quarter or more of the available directions to about a tenth, which we call subspace crystallization. (iii) Imposing low rank does not speed up grokking. Against a factorized full-rank control, capping the rank of the transformer's MLP input weights delays the onset, and capping the rank of its embedding stops 15 of 20 runs from grokking. Compression is therefore a signature of grokking rather than its cause.

Learning rate decay (LRD) is critical for training high-performance models, yet most designs remain empirically driven. Although the Universal Budget Aware (UBA) LRD offers a theoretical framework, its reliance on SGD dynamics limits its applicability to modern Adam-type optimizers and leaves its key `optimization difficulty parameter' opaque. To bridge this gap, we propose the Universal Training Adaptive (UTA) LRD. Unlike UBA's fixed offline trajectory anchored solely to iterations, UTA treats LRD as a closed-loop shape-and-scale problem. By continuous-time formulation, UTA is training-state adaptive with segment-wise optimal shape. This adaptivity links external iterations with internal training states, enabling a near-optimal path in both progression shape and learning rate (LR) scale, a property unattainable by other LRDs, including UBA. Empirically, we demonstrate consistent improvements across language models (LLaMA 60M to 1B) and vision models (ResNet50, ViT-B) on ImageNet with plug-and-play ease and negligible overhead. Extensive ablation studies further confirm UTA's robustness across dimensions, including peak LR, final decay ratio, its parameter $\varphi_0$, component efficacy and generalization to fine-tuning scenarios, underscoring its practical implementability.

Multilingual transformer models are often assumed to generalise across languages, but their effectiveness for coreference resolution in low-resource settings remains unverified. We investigate this gap in Bengali, a language with over 250 million speakers but limited annotated coreference resources. We evaluate transformer-based pre-trained encoders under multiple settings, including zero-shot, fine-tuning, data augmentation, and cross-lingual transfer. Our results challenge several common assumptions about model generalisation and data efficiency in low-resource NLP. We demonstrate that a small amount of high-quality annotated data significantly outperforms larger mixed-quality datasets (p < 0.01). We find that data augmentation, particularly back-translation, does not improve performance and can degrade results in coreference resolution. Cross-lingual transfer is also inconsistent: transfer from a distant language like English reduces performance, and transfer from the related language Hindi provides no consistent benefit across models. Several widely used multilingual models exhibit collapsed embedding spaces, rendering them unsuitable for similarity-based clustering tasks. These findings highlight the importance of annotation quality over data quantity and provide empirical guidance for developing coreference resolution systems in low-resource languages.

This article revisits entropy and related magnitudes (entropy density, effective measure complexity, and informational distance) for the analysis of sequential data. We present their mathematical foundations within information theory and ergodic theory, establishing connections between probabilistic Shannon entropy and algorithmic Kolmogorov complexity. A central result, Zivs theorem, demonstrates the asymptotic equivalence between the normalised number of distinct words and entropy density, providing the theoretical basis for estimation via Lempel-Ziv factorisation. We examine the convergence properties of these magnitudes and introduce estimation procedures for finite sequences, including a method for the effective measure complexity and the transfer entropy rate. An open-source software library implementing parallel algorithms for efficient Lempel-Ziv factorisation is presented. The practical utility of the framework is illustrated through three case studies: functional brain connectivity analysis from fMRI data, characterisation of spatiotemporal dynamics in coupled nonlinear oscillators, and electrocardiogram-based classification of atrial fibrillation, demonstrating how entropic measures capture meaningful structural and dynamical properties across diverse domains.

Generative AI offers new approaches to turbulence problems in which incomplete observations or unresolved d.o.f. admit multiple physically plausible solutions. Potential applications include stochastic closures, super-resolution, data assimilation, uncertainty quantification, and the generation of Eulerian and Lagrangian flow ensembles. However, turbulence is not a conventional image-generation problem. Its multiscale dynamics, intermittency, extreme events, rough velocity fields and intrinsically stochastic transport make visual realism, pointwise error, and low-order spectral agreement insufficient measures of physical fidelity. We examine where generative AI provide novel capabilities with emphasis on diffusion models, score-based methods, and related probability-transport formulations. We discuss their use in unconditional generation, probabilistic reconstruction, reduced-order modeling, temporal evolution, and mesh-free representations. We argue that validation must address turbulence properties: energy transfer, structure functions, anomalous scaling, pdf tails, coherent structures, conservation laws, temporal consistency, ensemble diversity and generalization. Finally, we discuss potential limitations in validation, baseline, data availability and reproducibility, and the need for a benchmark ecosystem combining open data, standardized challenges, hidden tests, physics-aware metrics, and documented failures. This paper serves as a bridge between two communities. For AI practitioners, we provide a fair and accessible introduction to turbulence, essential for appreciating the challenges posed by the field and for assessing the physical relevance of generative models. For turbulent practitioners, we provide an introduction to the key technological aspects of Generative AI, clarifying what these tools actually do, and where their potential and current limitations lie.

Multimodal video retrieval combines visual, acoustic, and speech information, yet how their fusion order should depend on the data and the retrieval query remains unclear. We introduce Unified Progressive Representation Learning for Order Analysis, or UniPRO, to investigate this question through configurable fusion schedules with matched model components and training settings. Starting from text-to-video retrieval, we find that the tested one-by-one schedules outperform stacked fusion, while changing the order within one-by-one fusion also changes performance. To guide order selection, we examine two complementary considerations: learning difficulty under unequal modality availability and preservation of information relevant to the query. Complete-modality controls show that missing inputs alone cannot explain order sensitivity. At R@1, text and audio retrieval favor schedules that introduce the modality matching the query last. Controlled speech missingness during training and testing reverses the preferred order in both R@1 and SumR, with different crossover points. These findings motivate balancing the two considerations when selecting a fusion path. We then use them to predict a visual-last schedule for image queries, which achieves 97.9 R@1, the highest among the evaluated orders on MSR-VTT-9k. Main experiments on four text-to-video benchmarks establish the effectiveness of UniPRO, with component ablations assessing its supporting design. Together, the results identify fusion order as a task-dependent choice that should account jointly for modality availability and the retrieval query.

Multivariate time series forecasting requires capturing temporal dependencies across multiple scales while accounting for both shared and variable-specific dynamics. Compact forecasting models have emerged as an attractive alternative to increasingly large architectures, but preserving strong predictive performance within tight computational and memory budgets remains challenging. We propose a compact one-layer convolutional framework that learns a shared bank of multi-scale temporal filters of each variable. Lightweight variable-specific scalar weights determine the contribution of each temporal component, enabling the model to capture shared temporal structure while adapting to variable-specific dynamics. The learned weights and filter-output energy further provide a natural signal for structured pruning of convolutional filters, reducing computational complexity with minimal degradation in forecasting performance. We additionally introduce an amplitude-adaptive normalization mechanism that preserves variable-dependent temporal amplitudes. Experiments on multiple datasets demonstrate forecasting performance competitive with state-of-the-art methods while requiring substantially fewer parameters and inference FLOPs and achieving low per-sample latency.

Large scale language model~(LLM) pretraining is increasingly bottlenecked by optimizer efficiency, driving optimizer design from vector-aware methods such as AdamW to matrix-aware methods. Matrix-aware optimizers such as Muon exploit matrix structure but treat layers independently, neglecting cross-layer correlations. Tensor-aware optimizers are therefore needed, yet existing ones rely on a matricization-based tensor algebra and remain restricted to shallow stacking and specific modules. In this paper, we propose \textbf{TUON}, a novel optimizer built on a transform-based tensor algebra derived from the Tensor-SVD (T-SVD). Unlike methods that polarize a globally unfolded matrix, TUON mixes layers via a unitary transform $\boldsymbol{Q}$ and then polarizes each transformed slice, and inverts the transform to recover the update. This mix-then-separate strategy captures cross-layer correlations while preserving per-layer diversity, avoiding the structural dimension imbalance and cross-layer misalignment of unfolding. Thus it successfully scales to more layers and diverse modules beyond QKV. Theoretically, we prove that TUON performs steepest descent under a $\boldsymbol{Q}$-transformed tensor spectral norm, and identify its update as the tensor polar factor of the T-SVD. We further establish transform non-redundancy, norm and smoothness comparisons with Muon and TEON, and a convergence guarantee. Empirically, TUON improves consistently over Muon and TEON on GPT-124M/350M and Qwen-0.6B/4B, with ablations confirming the transformation mechanism, cross-layer structure beyond two-layer stacking, adaptability beyond QKV, and robustness to the polarization method.

We study the problem of submodular maximization under a matroid constraint. In this problem, there is a monotone submodular function $f: 2^{N} \rightarrow \mathbb{R}^+$ and a matroid $M(N, I)$, accessible via value and independence oracles and the goal is to find the independent set $S \in I$ maximizing the objective function, i.e., $f(S) = \max_{I \in I} f(I)$. In the fully dynamic setting, the active ground set $N \subseteq N$ evolves through element insertions and deletions. The goal is to maintain a solution $S \subseteq N$ with $S \in I$ and $f(S) = \max_{I \subseteq N,, I \in I} f(I)$ after each update. Two approximation algorithms have been proposed to solve this problem: Dütting et al. developed a $(4+ε)$-approximation algorithm with an expected amortized update time of $O\!\bigl(\frac{k^2}ε \log k \log n \log^3 \frac{k}ε\bigr)$, and Banihashem et al. achieved a similar approximation with a better update time $O\!\bigl(k \log k \log^3 \frac{k}ε\bigr)$. However, these methods both become inefficient when the rank $k$ of the matroid is large (e.g., when $k = n^{o(1)}$). The central open question in this line of work is whether a fully dynamic algorithm with polylogarithmic update time exists. We resolve this question by presenting the first fully dynamic algorithm for submodular maximization over matroids with polylogarithmic update time. Our algorithm maintains a $(4+ε)$-approximate solution with expected amortized update time $poly(\log n, ε^{-1})$, where $n = |N|$. In particular, the update time is independent of the matroid rank $k$, thereby eliminating the dependence on $k$ and settling the open problem.